Weak solutions for forward--backward SDEs--a martingale problem approach

0.34 MB·English

by  Jin Ma

TheAnnalsofProbability
2008,Vol.36,No.6,2092–2125
DOI:10.1214/08-AOP0383
(cid:13)c InstituteofMathematicalStatistics,2008
WEAK SOLUTIONS FOR FORWARD–BACKWARD
SDES—A MARTINGALE PROBLEM APPROACH
9
0
By Jin Ma,1 Jianfeng Zhang2 and Ziyu Zheng3
0
2
University of Southern California and Barclays Capital

n
a In this paper, we propose a new notion of Forward–Backward
J
Martingale Problem (FBMP), and study its relationship with the

9 weak solution to the forward–backward stochastic differential equa-
1 tions(FBSDEs).TheFBMPextendstheideaofthewell-known(for-

ward) martingale problem of Stroock and Varadhan, but it is struc-
]
R tured specifically to fit the nature of an FBSDE. We first prove a
general sufficient condition for the existence of the solution to the
P
FBMP.IntheMarkoviancasewithuniformlycontinuouscoefficients,
.
h weshowthattheweaksolutiontotheFBSDE(orequivalently,theso-
at lutiontotheFBMP)doesexist.Moreover,weprovethattheunique-
m ness of the FBMP (whence the uniqueness of the weak solution) is
determined by the uniqueness of the viscosity solution of the corre-
[
sponding quasilinear PDE.

1
v
0 1. Introduction. The theory of backward stochastic differential equa-
9 tions (BSDEs for short) has matured tremendously since the seminal work
7
of Pardoux and Peng [24]. The fundamental well-posedness of BSDEs with
2
. various conditions on the coefficients as well as terminal conditions have
1
been studied extensively, which can be found in a large amount of litera-
0
9 ture. A commonly used list of reference include the books of El Karoui and
0 Mazliak [9] and Ma and Yong [18] for the basic theory of BSDEs, and the
:
v survey paper of El Karoui, Peng and Quenez [10] for the applications of
Xi BSDEs to mathematical finance. Itis worth noting that almost all the exist-
ing works on BSDEs or its extension Forward–Backward SDEs (FBSDEs)
r
a are exclusively considered in the realm of “strong solutions,” and a missing
Received February 2006; revised June 2007.
1Supportedin part by NSFGrants 05-05427 and 08-06017.
2Supportedin part by NSFGrant 04-03575 and 06-31366.
3Supportedin part by NSFGrant 03-06233.
AMS 2000 subject classifications. Primary 60H10; secondary 35K55, 60H30.
Key words and phrases. Forward–backward stochastic differential equations, weak so-
lutions, martingale problems, viscosity solutions, uniqueness.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Probability,
2008,Vol. 36, No. 6, 2092–2125. This reprint differs from the original in
pagination and typographic detail.
1
2 J. MA,J. ZHANGAND Z. ZHENG
piece of puzzle in the theory of BSDEs seems to have been the notion of
“weak-solution.” Such a notion, although extremely conceivable and tempt-
ing from both theoretical and practical point of views, has not been fully
explored.
In a recent paper, Antonelli and Ma [1] introduced the notion of weak
solution to a class of FBSDEs. In that paper, it was shown that some stan-
dard results regarding the relations among weak solution, strong solution
and different types of uniqueness still hold. However, the results in that pa-
per were a far cry from a systematic study for weak solutions. In particular,
theauthorswerenotabletoaddressthecoreissueregardingtheuniqueness.
Similar topics were studied later by Buckdahn, Engelbert and Rascanu [4],
with a more general definition and more extended investigation. But the
issue of uniqueness remains. Independent of our work, recently Delarue and
Guatteri [8] established the existence and uniqueness of weak solutions for
a class of Markovian FBSDEs by using elegantly the decoupling strategy
in the Four Step Scheme (cf. [16]). However, they require the coefficients
to be Lipschitz continuous in the backward components, and thus roughly
speaking their FBSDE is weak only in the forward component. To our best
knowledge, so far there has not been any work trying to address the issue
of uniqueness in law for a true BSDE/FBSDE; and it is our hope that this
paper could be the first step in that direction.
Our first goal of this paper is to findan appropriate definition of a “back-
ward” version of the martingale problem associated to the weak solution.
We shall follow naturally the idea of the forward martingale problem ini-
tiated by Stroock and Varadhan (cf., e.g., [26]), and recast the FBSDE in
termsofsomefundamentalmartingales,whichthenleadstothenotionofthe
Forward–Backward Martingale Problem (FBMP). Suchanotion extendsthe
usual martingale problem and it is equivalent to the weak solution defined
in [1]. Our objective then is to prove the existence and uniqueness of the
solution to FBMP, whence those of weak solution. Given the large amount
of recent studies on the existence of (strong) solutions to BSDEs/FBSDEs
withless-regular coefficients,notably theworksof[3,6,14]and[15],tomen-
tion a few, we are particularly interested in finding a unified method that
works for high dimensional FBSDEs with nonsmooth coefficients for which
a strong solution is less likely to exist. We shall first prove a general suffi-
cient condition for the existence of solution to FBMP, using mainly some
weak convergence arguments under Meyer–Zheng topology. A key element
in the sufficient condition could essentially be understood as a certain type
of tightness criterion for processes with paths in an L2 space, which shall
be further explored in our future publications. We will then show that such
sufficient condition can be verified in a Markovian case assuming that all
the coefficients are bounded and uniformly continuous.
WEAK SOLUTIONSFORFORWARD–BACKWARDSDES 3
The last part of this paper deals with the main issue: the uniqueness of
the solution to the FBMP. We note that to date the main difficulties in
the discussion has always been the martingale integrand in the BSDE (the
process Z), because in general one does not have a workable canonical space
for this process. In fact, although in many cases the process Z is c`adla`g or
even continuous (see, e.g., [20]), such path regularity is by no means clear a
priori. However, it is noted that if all the coefficients are Ho¨lder continuous,
one can show that the martingale integrand can be treated as a function
of the forward components of the solution, owing to the idea of the Four
Step Scheme of [16]. This fact, together with the procedure we used to
prove the existence, shows that at least one weak solution can bebuilt using
only the path spaces of the continuous components of the solution. This
result becomes more significant when we establish the uniqueness, since
it essentially eliminated the subtlety caused by the canonical spaces. Our
uniqueness proof is originated from the idea of “method of optimal control”
for solving an FBSDE (see [17, 18]). Although it may notbeintuitive dueto
the technicalities involved, the basic idea is to investigate a variation of the
notion of “nodal set” in [17], so as to show that uniqueness of the viscosity
solutions to the corresponding quasilinear PDE implies the uniqueness of
the solution to the FBMP (whence the weak solution). We should note that
inthispaperwearestillnotabletoprovetheuniquenessinthemostgeneral
sense, but we believe that our method has a potential to be applied to more
general FBSDEs, and the uniqueness should hold in a much wider class
of weak solutions. We hope to be able to address the issue in our future
publications.
The rest of the paper is organized as follows. In Section 2, we give the
preliminaries, recall the definition of weak solution and introduce the notion
of an FBMP. In Section 3, we prove the general sufficient condition for the
existence of the solution to FBMP. In Section 4, we consider the Markovian
case. Finally, inSection 5,weprovetheuniquenessof thesolution toFBMP.
2. Preliminaries. In this section, we give the basic probabilistic set up,
recall the definition of weak solution of an FBSDE and introduce the notion
of Forward–Backward Martingale Problems (FBMPs).
For any Euclidean space Rk, regardless of its dimension, we denote its
norm by . We denote C([0,T];Rk) to be the space of all Rk-valued con-
tinuous fu|n·c|tions endowed with the sup-norm; and D([0,T];Rk) to be the
space of all E-valued c`adla`g functions endowed with the Skorohod topology
(see, e.g., [11]). When k=1, we may omit R in the notation.
Foragivenfinitetimehorizon[0,T],wesaythataquintuple(Ω, ,P,F,W)
F
isastandard set-upif(Ω, ,P)isacompleteprobabilityspace;F=△ t t [0,T]
is a filtration satisfying tFhe usual hypotheses (see, e.g., [25]); and{WF }is∈an
4 J. MA,J. ZHANGAND Z. ZHENG
-Brownian motion. Inparticular,if = W,thenaturalfiltration gen-
{Ft} Ft Ft
erated by the Brownian motion W, augmented by all the P-null sets of
F
and satisfying the usual hypotheses, then we say that the standard set-up
is Brownian.
A.Weak solution of FBSDEs.Letusconsiderthefollowing forward–
backward SDE:
t t
X =x+ b(s,(X) ,Y ,Z )ds+ σ(s,(X) ,Y ,Z )dW ,
t s s s s s s s
(2.1)  Z0 Z0
T T
Yt=g((X)T)+ h(s,(X)s,Ys,Zs)ds ZsdWs.
−
Zt Zt
Here, (X ,Y ,Z ,W ) Rn Rm Rm d Rd, and the functions b, h, σ and
t t t t ×
∈ × × ×
g are functions with appropriate dimensions. We note, in particular, that
the coefficient b is a progressively measurable function defined on [0,T]
C([0,T],Rn) Rm Rm d with valued in Rn, and (X) denotes the path o×f
× t
X up to time×t. M×ore precisely, for each t [0,T] and (y,z) Rm Rm d,
×
∈ ∈ ×
the mapping x b(t,(x) ,y,z) is measurable with respect to the σ-field
t
7→
t(C([0,T];Rn)),where t(C([0,T];Rn))=△σ x(t ):x C([0,T];Rn) (cf.,
B B { ∧· ∈ }
e.g., [13]). Thecoefficients σ, h and g shouldbeunderstoodin asimilar way.
It is known that (cf., e.g., [18]) an adapted (strong) solution to the FBSDE
(2.1) is usually understood as a triplet of processes (X,Y,Z) defined on any
given Brownian set-up such that (2.1) holds P-almost surely. The following
definition of weak solution is proposed in [1].
Definition2.1. Astandardset-up(Ω, ,P, ,W)alongwithatriplet
t
F {F }
of processes (X,Y,Z) definedon this set-up is called a weak solution of (2.1)
if:
(i) the processes X,Y are continuous, and all processes X, Y, Z are
-adapted;
t
F
(ii) denoting f =f(t,(X) ,Y ,Z ) for f =b,σ,h, it holds that
t t t t
T
P (b + σ 2+ h + Z 2)ds+ g((X) ) < =1.
t t t t T
| | | | | | | | | | ∞
(cid:26)Z0 (cid:27)
(iii) (X,Y,Z) verifies (2.1) P-a.s.
We remark here that unlike a “strong solution,” a weak solution relaxed
the most fundamental requirement for a BSDE, that is, the set-up be Brow-
nian. But instead, it requires the flexibility of the set-up for each solution,
similar to the forward SDE case. We should point out that in [1] it is shown
that the weak solution of FBSDE (2.1) exists under very mild conditions,
and that there does exist a weak solution that is not “strong.”
WEAK SOLUTIONSFORFORWARD–BACKWARDSDES 5
Remark 2.2. Although in the basic setting of FBSDE (2.1), the co-
efficients are seemingly “deterministic,” it can be easily extended to the
“random coefficients” case. For instance, if we add the canonical Brownian
motionW intotheequation,andconsider(W,X)astheforwardcomponent,
then we can allow the coefficients to have the form
(2.2) f(t,ω,(X)t,YtZt)=△f(t,(W)t,(X)t,Yt,Zt), f =b,σ,h,
and the FBSDE (2.1) has nonanticipating random coefficients. In fact, our
general existence result Theorem 3.1 holds true for general FBSDEs with
coefficients of the form (2.2). However, at this stage, we feel that it is more
convenient to consider (2.1) in the given form so as to avoid further com-
plication in the proof of the uniqueness. We should note that even in the
standard (forward) martingale problem (cf. [26]), the component W is not
involved directly. (cid:3)
B. Forward–backward martingale problem. Before we define the
martingaleproblem,letusgiveadetaileddescriptionofa“canonical set-up”
on which our discussion will be carried out. Define
(2.3) Ω1=△C([0,T];Rn); Ω2=△C([0,T];Rm); Ω=△Ω1 Ω2,
×
where Ω1 denotes the path space of the forward component X and Ω2 the
path space of the backward component Y of the FBSDE, respectively.
Next, wedefinethecanonical filtration by Ft=△Ft1⊗Ft2,0≤t≤T, where
FtiI=△n σw{hωait(rfo∧llotw):sr,≥we0}d,enio=te1,t2h.eWgeendereincoetleemFe=△ntFoTf ΩanbdyFω=△={(Fωt1},0ω≤2t)≤,Ta.nd
denote the canonical processes on (Ω, ) by
F
xt(ω)=△ω1(t) and yt(ω)=△ω2(t), t 0.
≥
Finally, let (Ω) beall the probability measures definedon (Ω, ), endowed
P F
with the Prohorov metric.
To simplify presentation, we first assume that σ = σ(t,(x) ,y). Here,
t
we abuse the notation x by denoting elements of C([0,T),Rn) instead of
the canonical process. [The case σ =σ(t,(x) ,y,z) is a little more compli-
t
cated; we address it in Remark 2.4 below.] Further, for f =b, h, we denote
fˆ(t,(x) ,y,z)=f(t,(x) ,y,zσ(t,(x) ,y)), and let a=σσT. We give the fol-
t t t
lowing definition for a forward–backward martingale problem.
Definition 2.3. Let b, σ, h and g be given. For any x Rn, a solu-
∈
tiontotheforward–backwardmartingaleproblemwithcoefficients (b,σ,h,g)
[FBMP (b,σ,h,g) for short] is a pair (P,z), where P (Ω), and z is
x,T
a Rm n-valued predictable process defined on the filtered∈ Pcanonical space
×
(Ω, ,F), such that following properties hold:
F
6 J. MA,J. ZHANGAND Z. ZHENG
(i) the processes
t
Mx(t)=△xt ˆb(r,(x)r,yr,zr)dr and
−
Z0
(2.4)
t
My(t)=△yt+ hˆ(r,(x)r,yr,zr)dr
Z0
are both (P,F)-martingales for t [0,T];
∈
(ii) [Mi,Mj](t)= ta (r,(x) ,y )dr, t [0,T], i,j=1,...,n;
x x 0 ij r r ∈
(iii) M (t)= tz dM (r), t [0,T].
(iv) P yx =x 0=r1Ranxd P y ∈=g((x) ) =1.
{ 0 R} { T T }
We note that by (iii) we imply that the quadratic variation of M is
y
absolutely continuous with respect to the quadratic variation of M , thus in
x
the definition we require implicitly
T
P z a(t,(x) ,y )zT 2 dt< =1.
| t t t t |Rm×m ∞
(cid:26)Z0 (cid:27)
Remark 2.4. The case when σ =σ(t,(x) ,y,z) can be treated along
t
the lines of the “Four Step Scheme” (see, e.g., [16]). That is, one should
first find a function Φ:[0,T] C([0,T],Rn) Rm Rm n Rm d such
× ×
× × × 7→
that Φ(t,(x) ,y,z)= zσ(t,(x) ,y,Φ(t,(x) ,y,z)), and consider σ(t,(x) ,y,
t t t t
Φ(t,(x) ,y,z)). Then we define the forward–backward martingale problem
t
thesamewayasDefinition2.3exceptthatthefunctionsˆbandhˆ arereplaced
by fˆ(t,(x) ,y,z)=f(t,(x) ,y,Φ(t,(x) ,y,z)), f =b, h. We leave the details
t t t
to the interested reader.
We note that the Definition 2.3 looks slightly different from that of the
traditional martingale problem. But one can check that they are essentially
thesame,moduloanapplicationofItˆo’sformula.Infact,if(P,z)isasolution
to the FBMP (b,σ,h,g), then by Definition 2.3(i) and (iii), we have
x,T
dx =b(t,(x) ,y ,z )dt+dM (t),
t t t t x
(2.5) dy = h(t,(x) ,y ,z )dt+dM (t)
 t t t t y
 =b−h(t,(x) ,y ,z )dt+z dM (t).
t t t t x
−b
ApplyingItˆo’sformulaandusingDefinition2.3(ii),foranyϕ C2(Rn Rm)
b ∈ ×
and t [0,T], one has
∈
dϕ(x ,y )= ϕ(x ,y ),b(t,(x) ,y ,z )
t t x t t t t t
{h∇ i
ϕ((x) ,y ),h(t,(x) ,y ,z )
y t bt t t t
−h∇ i
+ 1tr D2 ϕ(x ,y )A(t,(x) ,y ,z ) dt
2 { x,y t bt t t t }}
+ ϕ(x ,y ),dM (t) + ϕ(x ,y ),dM (t) ,
x t t x y t t y
h∇ i h∇ i
WEAK SOLUTIONSFORFORWARD–BACKWARDSDES 7
where
I
A(t,(x)t,y,z)=△ zn a(t,(x)t,y)[In,zT];
(cid:20) (cid:21)
(2.6)
∂2 ϕ ∂2 ϕ
D2 ϕ= xx xy .
x,y ∂2 ϕ ∂2 ϕ
(cid:20) xy yy (cid:21)
Now, if we define a differential operator
Lt,x,y,z =△ 21tr{A(t,(x)t,y,z)Dx2,y}
(2.7)
+ b(t,(x) ,y,z), h(t,(x) ,y,z), ,
t x t y
h ∇ i−h ∇ i
then the fact that the (P,z) is a solution to the FBMP (b,σ,h,g) implies
b b x,T
that
t
(2.8) C[ϕ](t)=△ϕ(xt,yt)−ϕ(x,y0)− Ls,(x)s,ys,zsϕ(xs,ys)ds
Z0
is a P-martingale for all ϕ C2(Rn Rm). Conversely, if (2.8) is a P-
martingaleforallϕ C2(Rn ∈Rm),the×nwecanchooseappropriatefunction
∈ ×
ϕ sothatDefinition 2.3holds.Inother words,Definition 2.3actually reflects
all the necessary information for a “martingale problem.” But we prefer this
particular form as it is more symmetric and reflects the structure of our
FBSDE more explicitly.
The following theorem exhibits the connection between the weak solution
and the solution to the forward–backward martingale problem.
Theorem 2.5. Assume n=d. Assume also that σ=σ(t,(x) ,y) is non-
t
degenerate. ThenFBSDE(2.1)hasaweaksolutionifandonlyifFBMP (b,
x,T
σ,h,g) has a solution.
Proof. FirstassumeFBSDE(2.1)hasaweaksolution(X,Y,Z)defined
on a standard set-up (Ω, ,P, ,W). Note that
t
F {F }
t
[X,Y] = σ(s,(X) ,Y )ZT ds.
t s s s
Z0
Thus,sinceσ 1 exists,weseethatZ isadaptedto X,Y,thefiltrationgener-
−
F
ated by (X,Y).Usingtheforwardequation in(2.1),wecan furtherconclude
that W is also X,Y-adapted. Therefore, without loss of generality, we may
consider the caFnonical space Ω defined by (2.3), and let P=P (X,Y) 1
−
◦
be the distribution of (X,Y), so that (X,Y) is the canonical processes. De-
fine zt=△Ztσ−1(t,(x)t,yt). One can check straightforwardly that (P,z) is a
solution to FBMP (b,σ,h,g).
x,T
8 J. MA,J. ZHANGAND Z. ZHENG
We next assume FBMP (b,σ,h,g) has a solution (P,z). Define
x,T
t
(2.9) Wt=△ σ−1(s,(x)s,ys)dMx(s).
Z0
Then W is a continuous local martingale and [W,W] = t by definition.
t
Therefore, it follows from the L´evy characterization theorem (cf., e.g., [13])
we know that W is a Brownian motion. Now define Zt =△ ztσ(t,(x)t,yt).
One can easily check that (x,y,z,W), together with the canonical space, is
a weak solution to FBSDE (2.1). (cid:3)
Remark 2.6. (i) From theproofof Theorem2.5 weseethat theprocess
z in Definition 2.3 is different from the martingale integrand Z in FBSDE
(2.1). In fact, one has the relation: Z =z σ(t,(x) ,y ). Note that in the
t t t t
Markovian strong solution case the process z is actually associated directly
to the gradient of the solutions to a quasilinear parabolic PDE (see, e.g.,
[18]).
(ii) When σ is nondegenerate, there is an obvious one-to-one correspon-
dence between Z and z. Thus, we shall often refer to (P,Z) as a solution to
FBMP (b,σ,h,g) as well, when the context is clear. This is particularly
x,T
important in Section 5.
To conclude this section, let us give the following standing assumptions
which will be used in different combinations throughout the paper:
(H1) The coefficients (b,σ,h,g) are bounded, measurable functions, such
thatthemappings(x,y,z) f(t,(x) ,y,z),f =b,σ,h,g,and(x,y,z)
t
C([0,T];Rn) Rm Rm 7→n are uniformly continuous, uniformly in∈
×
× ×
t [0,T];
(H2) T∈hereexistsaconstantK >0,suchthat 1 λ 2 λTσσT(t,(x) ,y,z)λ
K λ 2, for all (t,x,y,z) [0,T] C([0,TK];|R|n)≤ Rm Rm nt and al≤l
×
λ | R| n; ∈ × × ×
∈
(H3) The mappings t f(t,(x) ,y,z), f =b, σ, h, and t [0,T] are uni-
t
formly continuou7→s, uniformly in (x,y,z) C([0,T];Rn∈) Rm Rm n.
×
∈ × ×
3. Existence: a general result. In this section, we study FBSDE (2.1).
We note thatin this section σ may dependon Z. Tosimplifypresentation in
whatfollows,weshallassumethatdim(X)=dim(Y)=dim(W)=1.Butwe
notethatallprocessesherecanbehigherdimensional,andallthearguments
can be validated without substantial difficulties. Denoting f =supf
to be the usual sup-norm of a (generic) continuous functiokn fk∞, our ma|in|
existence result is the following.
Theorem 3.1. Assume (H1), and assume that there exist a sequence
of coefficients (b ,σ ,h ,g ), n=1,2,..., such that:
n n n n
WEAK SOLUTIONSFORFORWARD–BACKWARDSDES 9
(i) for f =b,σ,h,g, f f 1;
(ii) all (b ,σ ,h ,g )k’snsa−tisfky∞(≤H1n), uniformly in n;
n n n n
(iii) for all n, the FBSDE (2.1) with coefficients (b ,σ ,h ,g ) have
n n n n
strong solutions (Xn,Yn,Zn), defined on a common filtered probability space
(Ω, ,P;F) with a given F-Brownian motion W;
F
(iv) denoting Zn,δ=△ 1 t Znds, it holds that
t δ (t δ)+ s
−
R T
(3.1) limsupE Zn Zn,δ 2dt =0.
δ→0 n (cid:26)Z0 | t − t | (cid:27)
Then (2.1) admits a weak solution in the sense of Definition 2.1.
Proof. We shall split the proof into several steps.
Step 1. Denote Θnt =△((Xn)t,Ytn,Ztn) and
t t t
Btn=△ bn(s,Θns)ds; Htn=△ hn(s,Θns)ds; An(t)=△ Zsnds;
Z0 Z0 Z0
t t
Mtn=△ σn(s,Θns)dWs; Ntn=△ ZsndWs.
Z0 Z0
Considerthesequenceofprocessesξn=(W,Xn,Yn,Bn,Hn,An,Mn,Nn),
n=1,2,...,anddefinethecanonicalspaceΩ=△D([0,T])8 withnaturalfiltra-
tion . Let Pn=△P[ξn]−1 (Ω) be the induced probability. It is fairly easy
F ∈P b
to show that all the components in the processes (W,Xn,Yn,Bn,Hn,An,
Mn,Nn) are quasimartingalesbwith uniformly bounded conditional varia-
tion. For example, let 0= t < <t = T be an arbitrary partition of
0 m
···
[0,T]. Then denoting Et=△E t , t 0, one has
{·|F } ≥
m 1
E − E Yn Yn + Yn
( | ti{ ti+1}− ti| | T |)
i=0
X
E m−1 ti+1 h (t,Θn) dt+ g (Xn) C.
≤ (i=0 Zti | n t | | n T |)≤
X
Here and in what follows, C >0 will denote a generic constant depending
onlyonthecoefficients (b,σ,h,g)andT,whichisallowedtovaryfromlineto
line. Thus, applying the Meyer–Zheng tightness criteria (Theorem 4 of [22])
we see that possibly along a subsequence, Pn converges to P (Ω) under
the Meyer–Zheng pseudo-path topology. Consequently, Pn co∈nPverges to P
weakly on D([0,T])8,andwedenotethelimittobe(W,X,Y,B,H,Ab,M,N).
Step 2. In the following steps, we shall identify the limit obtained in the
previousstep.Byaslightabuseofnotation,inwhatfollowslet(W,X,Y,B,H,
A,M,N)denotethecoordinateprocessofΩ.WefirstclaimthatP (W,X,Y,
{
b
10 J. MA,J. ZHANGAND Z. ZHENG
B,H,A) C([0,T])6 =1. Indeed, since by assumption (ii), all the coeffi-
∈ }
cientsareuniformlybounded,onecaneasilycheckthatthesequence (W,Xn,
Bn,Hn,An,Mn) is tight in the space C[0,T] under uniform topolo{gy. [For
}
ceoxnamtinpuleit,yifowfeMdnen,othteenwMitni(sδr)e=△adsiulyp|ss−eet|n≤δth|MatsnE−wMMtnn|(tδo)b2e tCheδ,muondiufolurms olyf
in n. Hence, by the standard tightness criteria on| the spa|ce≤ (C[0,T]) (see,
P
e.g., [2], Theorem 7.3), one can easily conclude that M is tight. Other
n
components can be argued similarly.] Consequently, t{he se}quence Pn re-
{ }
stricted to the components (W,X,Y,B,H,A) converges weakly to some
P˜ (C([0,T])6). Since C is a subspace of D, the uniqueness of the limit
th∈enPleads to that P˜=P , proving the claim.
(W,X,Y,B,H,A)
|
Next,byusingthedefinitionofweak convergence, itisfairlyeasytocheck
that
(3.2) X =X +B +M , Y =Y H +N t [0,T), P-a.s.
t 0 t t t 0 t t
− ∀ ∈
Clearly, under probability P, W is a Brownian motion. Since X,B,H,M
are all continuous, noting that sup E T Zn 2dt< , it follows from [22],
n 0 | t | ∞
Theorem 11, that M, N are both martingales. Further, applying [22], Theo-
R
rem10,weconcludethatAisabsolutelycontinuous,P-a.s.;andA = tZ ds
t 0 s
with EP T Z 2dt< .
0 | t| ∞ R
Step 3.WeshowthatB = tb(s,Θ )dsandH = th(s,Θ )ds, t,P-a.s.
R t 0 s t 0 s ∀
To this end, we note that the function b is uniformly continuous on z. Thus,
R R
for any ε>0, there exists ε >0 so that b(t,(x) ,y,z ) b(t,(x) ,y,z ) ε
0 t 1 t 2
| − |≤
whenever z z ε . Furthermore, (3.1), we can choose δ >0 such that
1 2 0 0
| − |≤
for any δ δ it holds that
0
≤
T
(3.3) supE Zn Zn,δ 2dt εε2.
n (cid:26)Z0 | t − t | (cid:27)≤ 0
assNumowptlieotnu(si)daenndotteheZdtδe=△fin1δi[tAiotn−ofAt−Pδn],,wohneereveAritfi=e△s0eafsoirlytt<ha0t. Then by
{ }
t
P
E B b(s,Θ )ds
t s
−
(cid:26)(cid:12) Z0 (cid:12)(cid:27)
(cid:12) (cid:12)
(cid:12)(cid:12)= limEP B t(cid:12)(cid:12)b(s,(X) ,Y ,Zδ)ds
δ→0 (cid:26)(cid:12) t−Z0 s s s (cid:12)(cid:27)
(cid:12) (cid:12)
(3.4) = lim lim (cid:12)(cid:12)EPn B tb(s,(X) ,Y ,Z(cid:12)(cid:12) δ)ds
δ→0n→∞ (cid:26)(cid:12) t−Z0 s s s (cid:12)(cid:27)
(cid:12) (cid:12)
(cid:12)t t (cid:12)
= lim lim E (cid:12) b (s,Θn)ds b(s,(Xn)(cid:12),Yn,Zn,δ)ds
δ→0n→∞ (cid:26)(cid:12)Z0 n s −Z0 s s s (cid:12)(cid:27)
(cid:12) (cid:12)
(cid:12) T (cid:12)
lim lim E (cid:12) b(s,(Xn) ,Yn,Zn) b(s,(Xn) ,Yn,Zn,δ(cid:12)) ds .
≤δ→0n→∞ (cid:26)Z0 | s s s − s s s | (cid:27)